Watermarking 3D mesh by spherical parameterization

Abstract

In this paper, a robust 3D trianglular mesh watermarking algorithm is presented by applying spherical parameterization. First, we transform the coordinate signals of the 3D triangular mesh into spherical signals using a global spherical parameterization and an even sampling scheme. Then, spherical harmonic transformation is used to generate some data for embedding watermarks. As a result, the watermarks can be embedded in the Fourier-frequency domain of the original mesh. Experimental results show that our watermarking algorithm is robust since watermarks can be extracted without mesh alignment or re-meshing under a variety of attacks, including noise addition, crop, filtering, enhancement, rotation, translation, scale and re-sampling.

Introduction

Currently, watermarking technology focuses on media types such as still images, audio and video streams [1], [2], [3], [4], [5], [6]. The problem of watermarking 3D models, on the other hand, has received less attention from researchers. However, as more and more CAD-based 3D data are entering the World Wide Web, it is inevitable that companies or copyright owners who present or sell their products in the virtual space will face copyright-related problems. A logical demand from these companies would be preventing their 3D-based material from unauthorized use. Digital watermarking will possess extensive application perspective in this area.

In comparison to audio, video, and still-image data, some challenges exist for watermarking 3D mesh models [7], [10]. For example, arbitrary meshes lack a natural parameterization for frequency-based signal processing, and thus they cannot adopt the corresponding image watermarking algorithms that are based on the frequency domain, such as the wavelet transforms and the Fourier transforms which most image watermarking algorithms are based on.

There are two kinds of 3D mesh watermarking algorithms that are similar to image watermarking. One is based on the spatial domain [7], [8], [9], which provides many insights into mesh watermarking but the techniques are not yet robust enough. For example, these algorithms cannot withstand the attacks of noise addition and cropping operations.

The other is on the “frequency” domain [10], [11], [12], [13], [14], [15]. Praun and Hoppe [10] modified the shape of the mesh by a spatial kernel to embed information in the “low-frequency” component of the shape. However, the watermarks are real numbers sampled from a Gaussian distribution. Kanai et al. [11] were the first to apply a transformed-domain watermarking approach on 3D meshes—a robust, blind-detection watermarking algorithm that works in the mesh's wavelet-transformed domain. However, the method requires the mesh to have 1–4 subdivision connectivity. Yin et al. [12] reported an informed-detection, robust mesh-watermarking algorithm that worked in a transformed domain. The algorithm is based on a multi-resolution decomposition of polygonal mesh shapes proposed by Guskov et al. [13]. However, registration and resampling are needed to bring the attacked mesh model back into its original location, orientation, scale, topology and resolution level when the watermarked mesh is cropped or similarity transformation. Recently, Ohbuchi et al. [16] presented an algorithm based on their previous works [14] in an attempt to watermark much larger meshes within a reasonable time. Although the watermarking is more robust against connectivity alteration, cropping, mesh simplification and smoothing, it still left some space for improvements. For example, watermark extraction requires a lot of information such as the original mesh, the patch key and the information that uniquely determines the partitioning of the original mesh. Mesh alignment and re-meshing are also needed during watermarks extraction.

In this paper, a new 3D mesh watermarking algorithm is developed using spherical parameterization. The spherical parameterization make the coordinate information of 3D meshes defined on a sphere in order to embed watermarks by spherical harmonic transform. The spherical harmonic transform is widely used in broad area of earth science, which really is the combination of Fourier transform in longitude and latitude transform in latitude [16]. Therefore, the spherical harmonic transform is introduced to analyze and synthesize spherical geometric signals, which made it possible to embed watermarks in the Fourier frequency domain of the original 3D mesh. So, our algorithm is robust to some attacks. However, these transformations can bring serious distortion to the watermarked mesh. Thus, we also use some methods to weaken the distortion, including reordering the vertices of the original mesh, modifying the watermarked 3D mesh to decrease the distortion during embedding watermarks, magnifying the watermark information of the 3D mesh during watermark extraction and coding watermarks in some particular ways.

As a result, our algorithm is proved to be strong to some attacks including noise addition, filtering, enhancement, rotation, translation and resampling without the need of mesh alignment and re-meshing. It is also robust to attacks of cropping, vertex reordering and simplification with preprocessing. In addition, only the spherical parameterization data of the original mesh is required while extracting watermarks.

In Section 2, the mesh watermarking algorithm is described in detail. Sections 3 present experimental results and attack analysis. In Section 4, we draw a conclusion.